What is the best way to reach the hotel from Heathrow and what would the approximate cost be.
It depends on how much you want to spend. We've used the hotel's Rolls Royce Phantom for a transfer a couple of times and it cost ~£500. A cheaper alternative would be Uber Lux which would cost around £70. Even cheaper would be the Tube or Heathrow Express.
Get an answer from 45 Park Lane - Dorchester Collection staff and past guests.